Schedule
Lecture Exp
Date
A
July 5
3
B
July 10
4
A
July 12
5
B
July 17
6
A
July 19
B
July 24
A
July 26
B
July 31
Lecture Topics
Course Overview
Discussion of Exp 1 – Goals, setup
(Deduce mean density of the
earth)
Measurements, uncertainties.
Statistical Analysis
Intro to error propagation
Discussion of Exp 2 – goals, setup
(Deduction of mass distribution)
Histograms & distributions
The Gaussian Distribution,
Maximum likelihood,
Rejected data, Weighted mean
Discussion of Exp 3 – goals, setup
(Tune a shock absorber)
Fitting
Chi-squared test of distribution
Discussion of Exp 4 – goals, setup
(Calibrate a voltmeter)
Chi-squared
Covariance and correlation
Final Exam Review
August 2
Final Exam
1
July 3
2
1
2
3
7
8
9
10
4
Assignment
Lab: -Prepare for Quiz #1
Taylor: -Read chapters 1-3, HW 1
Lab: -Analyze data for Exp #1
Taylor: -Read chapter 4, HW 2
Lab: -Prepare for quiz #2
Taylor: -Read chapter 5, HW 3
Lab: -Analyze data for Exp #2
Taylor: -Read chapters 6-7, HW 4
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
Taylor:
Lab:
8PM, York 2722
Physics 2BL Summer I 2012
-Prepare for quiz #3
-Read chapter 8, HW 5
-Analyze data for Exp #3
-Read chapters 9 & 12
-Prepare for quiz #4
-HW 6
-Analyze data for Exp #4
-Prepare for final exam
-Pick up graded work from
TAs
-Pick up final from LTAC
2nd Quiz
3rd Quiz
4th Quiz
1
Exp 2 – Goals, Setup
Histograms & Distributions, Limiting
Distribution, Normal Distribution
Lecture # 3
Physics 2BL
Summer Session I 2012
Physics 2BL Summer I 2012
2
Lecture #3:
• Issues from last week’s lab?
– Quiz #2 Thursday (complete prelab 2, HW 2-3)
•
•
•
•
Introduction to Experiment 2
Recap from last lecture
Distributions
Homework
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The Four Experiments
•
• Non-Destructive measurements of densities, inner
structure of objects
– Absolute measurements vs. Measurements of variability
– Measure moments of inertia
– Use repeated measurements to reduce random errors
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Experiment 2
• Measure the variation in thickness (due to
manufacturing) of the shell of a large
number of racquetballs in shipments
arriving at a number of stores, to determine
if the variation in thickness is much less
than 10%
Problem can be solved by
measuring the mass (M) and
moment of inertia (I) of the balls
NOTE: measuring the variation in
thickness, not the actual thickness
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Check the variation is
less than 10% using
a non-destructive
measurement!
Measure variation in
thickness by
measuring the
moment of inertia
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Summary of Exp 2
• Want to measure variation in thickness due to
manufacturing error only
• Thickness can be related to time through moment of
inertia: roll racquetballs down a ramp!
• Need to isolate manufacturing error by eliminating random
error:
– Set up ramp, perform calibration to estimate variation in time due
to random error – 1 ball N times
– Estimate total variation in time due to manufacturing error (AND
random error, since it is impossible to avoid) – N ball 1 time
• Calculate moment of inertia, I, from time, t
– Propagate error in t to get error in I
• Calculate variation in thickness due to manufacturing using
variation in time (independent of random error)
Estimate how many measurements are needed for the desired accuracy
Physics 2BL Summer I 2012
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Reminder – Moment of Inertia
• Rotational analog of mass (inertia of a rigid rotating body
with respect to its rotation axis)
I = ∫ r dm
2
• m is the mass, r is the distance of infinitesimal mass, dm to
the axis of rotation
• Units: kg m2
• Moment of inertia of a solid sphere about COM
I sphere
2
2
= MR
5
Outer radius
• Moment of inertia of hollow sphere about COM
5
5
2 R −r
Inner radius
I hollow =
5
M
R3 − r 3
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Exp 2 Setup
Racquetball
x
h1
Photo-gate timer
h2
• Parameters to measure once and use for all tests:
– Distance, x, b/n photogates (how to measure?)
– Starting and ending height above the table, h1 & h2, to
calculate the height difference, h, between the photo-gates
– Mass of ball, M (for 1 ball N times)
– Width of the groove, w, on the rail
– Outer radius of the ball, R
Calculate rolling radius, R′ = R 2 − (w 2 )2
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Calculate I
Racquetball
x
h1
Photo-gate timer
h2
1
1
2
2
Conservation of energy: Mgh = Mv f + Iω f
2
2
Rolling without slipping: v = R′ω
1 2
I 
Mgh = v f  M + 2 
2 
R′ 
h = h1 − h2
Kinematics for constant linear acceleration
0
v f = vi + at
0
x = vi t + 1 at 2
2
Assume starting from rest
2x
vf =
t
2 x2 
I 
Mgh = 2  M + 2 
t 
R′ 
2


ght
2
Solve for I: I = MR′  2 − 1
 2x
Physics 2BL Summer I 2012
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Repeating Measurements
• Take multiple measurements to estimate the error
in single time measurements (standard deviation)
σt =
1
2
(ti − t )
∑
N −1
• Take as many measurements as you can to reduce
the error in mean value of time (SDOM)
σt =
σt
N
– Photo-gate measures to 1 ms = 0.001 s
• As N increases, standard deviation will approach a
constant value, while SDOM will fall off as 1 N
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Measuring Variation due to Manufacturing
Chauvenet
σ measurement
σ total
σ manufacture
σ total = σ manufacture 2 + σ measurement 2
σt
Physics 2BL Summer I 2012
σI
σd
12
Propagate Error from d to I - Numerically
Moment of inertia of ball(s)
Prelab question 1
2
5
5


ght
2
R
−
r
I = MR′2  2 − 1 = M 3 3
 2x
 5 R − r To simplify our calculations
Dimensionless moment of inertia
~
I =
 2 1 − (r R )5
I
R′2  ght 2
= 2  2 − 1 =
3
2
MR
R  2x
5
(
)
1− r R

Typical values for the balls: d = 4.5 mm and R = 28.25 mm,
5
~ 2 1 − (23.75 28.25)
so r = R - d = 23.74 mm
I =
= 0.5717
3
5 1 − (23.75 28.25)
Small perturbation…
Take d = 4.4 mm, then r = 23.85 mm ~ 2 1 − (23.85 28.25)5
I =
= 0.5736
3
Fractional errors (variation in d):
5 1 − (23.85 28.25)
~
δd 4.5
4.5 − 4.4
4.4
δ
I
0.5736 − 0.5717
0.5717
0.5736
=
= 0.0222
=
= 0.00331
~
d
4
.
5
4.5
I
0.5736
0.5736
To know d to 10% need to know I to 1.5%
~
~
δd 0.0222 δI
δI
=
=
6
.
7
Need to relate fractional error in I to t…
~
~
Or:
d
0.00331 I
I
Physics 2BL Summer I 2012
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Propagate Error from I to t - Numerically
~
δt
δd
δI
= 6.7 ~ = (??)
t
d
I
Previous slide:
Dimensionless I in terms of t: I~ = 25 11 −− ((rr RR ))
~
~ ∂I
Error propagation formula: δI = ∂t δt

R′2  ght 2
= 2  2 − 1
R  2x

5
3
~
R′2 ght
~ ∂I
δI =
δt = 2 2 δt
∂t
R x
 ~ R′2  δt
~ R′2 ght 2 δt
= 2 I + 2 
δI = 2 2
R  t
R x t

Plugging in numbers
~
~
δI 2(I + R′ R ) 2(0.572 + R′
=
=
2
~
I
2
2
~
I
0.572
R′2 = R 2 − (w 2 )
2
R′2 R 2 = 1 − (w 2 ) R 2
δt
R 2 ) δt
≈4
t
t
2
Finally relate fractional error in d to t
~
δt
δt
≈ 6.7 ~ ≈ 6.7 ⋅ 4 ≈ 27
d
I
t
t
δd
δI
δd
d
≈ 27
δt
t
To obtain d with a precision of 10% need precision
of 0.4% in t
Physics 2BL Summer I 2012
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Histograms & Limiting Distributions
• Multiple, N, measurements of the same quantity, x
• Calculated “average” and “spread” of values
– Mean and std dev
• Determined the uncertainty of the mean
– SDOM
• Convenient way to visualize data: plot measured
value in binned histogram
• As N becomes very large, approach limiting
distribution
Physics 2BL Summer I 2012
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How to draw a Histogram
• Determine the range of your data
(largest value - smallest value)
• Choose number of bins ≥ 3 ≈ N
• Width of bins, ∆k, is range divided by #
• List bin boundaries, count number of data
points, nk, in each bin
• Draw histogram, y-scale, fk, may be the #
measurements in each bin
(
Physics 2BL Summer I 2012
)
16
Normalized Histogram
• Want the area of each bin to equal the
probability of finding a measurement within
that bin
∆k
• Area of rectangle: Ak = fk ∆k fk
A
k
– fk = vertical scale
– ∆k = width of bins
n1 n2 n3 n4 n5
N = n1 + n2 + n3 + n4 + n5
• Fraction of decays in bin: Fk = nk/N
– nk = # measurements in kth bin
– N = total number of measurements
• Choose fk so Ak = Fk
(Total area = 1)
Physics 2BL Summer I 2012
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Limiting Distribution
What fraction of data falls
within some range?
= ∫ f ( x )dx
Physics 2BL Summer I 2012
B
A
18
Limiting Distribution is Normalized
Need to have normalization in the limiting
distribution
∫
∞
−∞
f ( x )dx = 1
To be able to interpret:
∫
B
A
f ( x )dx
as the probability
Physics 2BL Summer I 2012
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The Guassian, or Normal Distribution
The limiting distribution for a measurement
subject to many small random errors is
bell shaped and centered on the true value
of x
The mathematical function that describes the
bell-shaped curve is called the normal
distribution or Gauss function
σ = width parameter
X = true value of x
Prototype function:
− ( x − X )2 2σ 2
e
Physics 2BL Summer I 2012
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The Gauss, or Normal Distribution
Normalize e
− ( x − X ) 2σ
2
2
∫
∞
−∞
f ( x )dx = 1
1
− ( x − X )2
G X ,σ ( x ) =
e
σ 2π
2σ 2
Width paramter σ ~ standard deviation, σx
True value X ~ the mean value of x
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σ
σ
σ
σ
σ
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0.6827
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Physics 2BL Summer I 2012
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Table
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Compatibility of measured result(s): t-score
t=
x A − xB
σ A2 + σ B 2
• What is a t-score:
– Ratio of discrepancy to error of the discrepancy
• Used to find probabilities, confidence levels
(probability of getting a worse result)
• Ex, Lab 2: compare one of your measured
values to the mean value of your data
– Best estimate of x: x ± σ x
– Single data point: x ± σ x
x−x
– t-score: t = x − x
=
σ x2 + σ x 2
σx
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Compatibility
t=
x−x
σx
• Probability of obtaining a value that differs from the mean
by t or more standard deviations is:
– Prob(outside tσ) = 1 – Prob(within tσ)
– AKA “confidence level”
• Large confidence level means likely outcome and hence
reasonable discrepancy
Define
<5% - significant discrepancy, t > 1.96
<1% - highly significant discrepancy, t > 2.58
boundary for unreasonable improbability
erf (t ) = ∫
X + tσ
X −tσ
G X ,σ ( x )dx
If the discrepancy is beyond the chosen boundary
for unreasonable probability, the theory and
measurement are incompatible (at the stated level)
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Rejection of Data
• Rejecting data can bias your measurements
– Never do so lightly!!!
• If there is reasonable suspicion of a mistake,
data should be rejected regardless of the
values measured
• If only the measured value is suspicious,
use Chauvenet’s criterion
– A suspicious data point is to be rejected if total
probability of measuring a worse result is less
than 50% (see Taylor)
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Rejection of Data – Chauvenet’s Criterion
Measure time (s): 3.8, 3.5, 3.9, 3.9, 3.4, 1.8 REJECT
1
1
2
x = ∑ xi = 3.4 s
(
)
σx =
x
−
x
= 0.8s
∑
i
N
N −1
xsus − x 1.8 − 3.4
tsus =
=
= 2.00 Prob(|t| ≤ 2.00) = 0.9545
σx
0.8
Prob(outside 2σ) = 1- Prob(within 2σ ) = 1 – 0.9545 = 0.0455
Total probability n = N * Prob(outside 2σ) = 6*0.0455 = 0.273
If n < 0.5, measurement is improbable and can be rejected
according to Chauvenet’s criterion!
x1 ,..., xN
tsus =
xsus − x
Chauvenet’s criterion
σx
n = N * Prob(|t| ≥ tsus)
If n < 0.5, the reject xsus
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Chauvenet’s Criterion
• If you are able to reject xsuspect, redo the
statistical analysis
• Previous question:
[3.092 s, 3.101 s, 3.098 s, 3.095 s, 4.056 s]
• New mean
3.092 + 3.101 + 3.098 + 3.095
t=
= 3.0965s
4
(Formerly 3.288 s)
• New standard deviation:
1  4
2
(
)
σt =
t
−
3
.
0965
= 0.00387s
∑
i


4 − 1  i =1

Physics 2BL Summer I 2012
(Formerly 0.4291 s)
33
Homework
Prelab problems for Exp 2
Read Taylor chapter 5
HW3: Taylor problems 5.2, 5.6, 5.20, 5.36
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